Write programme in C# and provide and explain Hoare's triple for this algorithm (pre-condition, post-condition and loop invariant):

Rearrange the elements in an integer array of size N so that all negative elements go first, then all zeros and then all positive elements in time O(N)

N. B. Use standard C# sorting and use only simply variables (i.e. no extra array, list, stack, hashset, etc.)